Problem 45 - Applied Calculus

Question

A person's blood pressure, \(p\), in millimeters of mercury \((\mathrm{mm} \mathrm{Hg})\) is given, for \(t\) in seconds, by $$p=100+20 \sin (2.5 \pi t)$$ (a) What are the maximum and minimum values of blood pressure? (b) What is the interval between successive maxima? (c) Show your answers on a graph of blood pressure against time.

Step-by-Step Solution

Answer

(a) Max: 120 mmHg, Min: 80 mmHg. (b) 0.8 seconds. (c) Graph oscillates between 80 and 120 mmHg.

Step 1 - Understanding the Function

The blood pressure, \( p(t) \), is given by the function\( p(t) = 100 + 20 \sin(2.5\pi t) \). This function has the form \( A + B\sin(Ct) \) where the amplitude is \( B = 20 \), the mean value is \( A = 100 \), and \( C = 2.5\pi \).

Step 2 - Calculating Maximum and Minimum Values

For a sine function \( \sin(x) \), the maximum value is \( 1 \) and the minimum is \( -1 \). Therefore, the maximum value of \( p(t) = 100 + 20 \sin(2.5\pi t) \) is \( 100 + 20 \times 1 = 120 \). The minimum value is \( 100 + 20 \times (-1) = 80 \).

Step 3 - Finding the Interval Between Successive Maxima

The period of the sine function \( \sin(2.5\pi t) \) is \( \frac{2\pi}{2.5\pi} = \frac{2}{2.5} = \frac{4}{5} \) seconds. Since the sine function reaches its maximum value once every period, the interval between successive maxima of \( p(t) \) is \( \frac{4}{5} \) seconds.

Step 4 - Sketching the Graph

To sketch the graph, plot time \( t \) on the x-axis and blood pressure \( p \) on the y-axis. The graph oscillates between 80 and 120 mmHg, with peaks at 120 mmHg occurring every \( \frac{4}{5} \) seconds. The midline of the oscillation is at 100 mmHg.

Key Concepts

Amplitude and Period

Understanding the amplitude and period in trigonometric functions is crucial for analyzing oscillations, like those in a sine wave.
This knowledge is particularly useful when dealing with real-world applications, such as modeling blood pressure.

Amplitude measures the extent of oscillation. In the function \( p(t) = 100 + 20 \sin(2.5\pi t) \), the amplitude is 20. This indicates how far the blood pressure deviates from its average value in either direction. - The higher the amplitude, the greater the variation above and below the mean value. - Here, the blood pressure can oscillate 20 mmHg above or below the average value of 100 mmHg.

The period of a sine function represents the time it takes for one complete cycle.
The period of \( \sin(2.5\pi t) \) is calculated using the formula \( \frac{2\pi}{C} \), where \( C \) is the coefficient of \( t \).

In this example, the period is \( \frac{2\pi}{2.5\pi} = \frac{4}{5} \) seconds. This tells us that every \( \frac{4}{5} \) seconds, the blood pressure completes a full oscillation.
Understanding how these elements interact allows for more accurate predictions and interpretations of repeated patterns in scenarios like blood pressure monitoring.

Graphing Sine Functions

Drawing a sine function graph helps visualize the behavior of oscillating phenomena such as blood pressure over time.
For the function \( p(t) = 100 + 20 \sin(2.5\pi t) \), we need to consider the parameters: amplitude, period, and midline.

To graph this function:

• Identify the midline of the graph, which is constant at 100 mmHg, around which the sine wave oscillates.
• Determine the amplitude, which sets the peak at 120 mmHg and trough at 80 mmHg.
• Find the period, computed to be \( \frac{4}{5} \) seconds.

Then, plot the time \( t \) on the x-axis and the blood pressure \( p \) in mmHg on the y-axis.
This lets us see the wave oscillating between the amplitude extremes, peaking at specified intervals.

Such graphs provide an intuitive view of how blood pressure varies over time, reflecting cyclical changes.
Key points to mark include the function's peaks, troughs, and midline intersections.
Practical graphing aids in predictive modeling, which is vital for medical monitoring and analysis.

Blood Pressure Modeling

Modeling blood pressure using trigonometric functions is a powerful technique that reflects the periodic nature of physiological processes.

In the given example, the function \( p(t) = 100 + 20 \sin(2.5\pi t) \) is employed to depict blood pressure as oscillations.- The mean blood pressure is 100 mmHg.- The amplitude of 20 mmHg indicates normal cyclical variation.

This model can help visualize how blood pressure rises and falls naturally, factoring in both amplitude and periodicity.
Real-world uses include identifying blood pressure trends that signal health conditions.

By adjusting parameters in the function (such as amplitude), different individual profiles can be modeled, aiding in personalized medical interventions.
Technology can employ mathematical models like this to ensure constant health monitoring.- These models offer valuable insights for healthcare professionals, enabling personalized care through precise, data-driven insights.
Using trigonometric models in this context offers both predictive and diagnostic power, essential in increasing effectiveness in healthcare solutions.